Mathematical Institute, Oxford

Let $X$ be a smooth hypersurface in projective space over a field $K$ of characteristic zero and let $U$ denote the open complement. Then the elements of the algebraic de Rham cohomology group $H_{dR}^n(U/K)$ can be represented by $n$-forms of the form $Q \Omega / P^k$ for homogeneous polynomials $Q$ and integer pole orders $k$, where $\Omega$ is some fixed $n$-form. The problem of finding a unique representative is computationally intensive and typically based on the pre-computation of a Groebner basis. I will present a more direct approach based on elementary linear algebra. As presented, the method will apply to diagonal hypersurfaces, but it will clear that it also applies to families of projective hypersurfaces containing a diagonal fibre. Moreover, with minor modifications the method is applicable to larger classes of smooth projective hypersurfaces.

Suppose that $C$ and $C'$ are cubic forms in at least 19 variables over a

$p$-adic field $k$. A special case of a conjecture of Artin is that the

forms $C$ and $C'$ have a common zero over $k$. While the conjecture of

Artin is false in general, we try to argue that, in this case, it is

(almost) correct! This is still work in progress (joint with

Heath-Brown), so do not expect a full answer.

As a historical note, some cases of Artin's conjecture for certain

hypersurfaces are known. Moreover, Jahan analyzed the case of the

simultaneous vanishing of a cubic and a quadratic form. The approach

we follow is closely based on Jahan's approach, thus there might be

some overlap between his talk and this one. My talk will anyway be

self-contained, so I will repeat everything that I need that might

have already been said in Jahan's talk.

Suppose a power series $f(x):= \sum_{n=0}^{\infty} a_{n} x^{n}$ has radius of convergence equal to $1$ and that $lim_{x\rightarrow 1}f(x) = s$. Does it therefore follow that $\sum_{n=0}^{\infty} a_{n} = s$? Tauber's Theorem answers in the affirmative, \textit{if} one imposes a certain growth condition (a \textit{Tauberian Condition}) on the coefficients $a_{n}$. Without such a condition it is clear that this cannot be true in general - take, for example, $f(x) = \sum_{n=0}^{\infty} (-1)^{n} x^{n}.$